Numerical Methods for a Reduced Model in Thin-Film Micromagnetics

Abstract

<p>We address the conformal finite element approximation to a reduced 2-d model arising in thin-film micromagnetics. The model was derived by DeSimone, Kohn, Müller, and Otto for a thin-film ferromagnetic element under an external magnetic field. The micromagnetic energy of the in-plane magnetization consists of two contributions: the energy of the stray field, and the Zeeman energy. The magnetization itself has to satisfy a convex constraint. We consider the approximation of the magnetization by Raviart-Thomas elements, which fits naturally in the theory of discrete de Rham complexes, a common concept of computational electromagnetism. The numerical challenge in micromagnetic simulations is the determination of the stray field, which in our case amounts to the evaluation of the single layer potential operator. We apply the method of hierarchical matrices developed by Hackbusch et al. to carry out calculations in sub-quadratic time. For weak external fields the problem for the magnetostatic charge distribution is a variational formulation of the Dirichlet screen problem. The charge distribution is known to have characteristic singularities near the edges and corners of the thin film cross-section. We establish the minimal regularity theory required later. We establish an a-priori error estimate in the energy norm and derive the necessary refinement rule for triangulations to retain the optimal rate of convergence. When the convex constraint on the magnetization becomes active, we apply an interior point method to compute an energy minimizing magnetization. A logarithmic barrier leads to a well-characterized, unique minimizer, the analytic center, though the energy is highly degenerate. Then we consider how to construct minimizers close to unit length: these correspond to saturated magnetizations observed in physical experiments. We apply modifications of well-established numerical schemes for the computation of viscosity solutions to Hamilton-Jacobi equations. We confront our numerical simulations with pictures from physical experiments, kindly provided by R. Schäfer. </p>